From a technical standpoint I don't see how there's much of a difference, given that devices are almost always connected to the Internet and almost all chat, messaging, and email services people use have a third party server that persists a message regardless of the recipient's "presence" at the time the message is sent. These services have converged, apart from a few idiosyncrasies and perhaps some waning differences in the social expectations of a particular service.
AIM added offline messages at some point, although I don't remember when. ICQ had it from the beginning (I think originally the client may have polled the server rather than running a persistent connection?, so the server had to store stuff a bit anyway; clients would do p2p messaging if both were online and could connect, otherwise would fall back to sending messages through the server)
According to my memory AIM didn't support that in the late 90s and early 2000s, and that was a key advantage of ICQ. But I mean in the last 15 years or so, corresponding roughly with the rise of ubiquitous smartphones (and foreshadowed by cell-connected PDAs and some feature phones).
